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Hi everybody,

This is my first post in this forum and first participation to the quest for the MandelGrail.

Recently, I tried to see how changing the coefficients of a quadratic "multiplication" could modify the shape of a Mandelbrot 3D.

Here is the iteration function z^2 + c :
x = v * mx * v + cx
y = v * my * v + cy
z = v * mz * v + cz
where v = (x, y, z) and mx, my and mz are 3x3 symmetrical matrices, with coefficients mxij . So, this can be written :
x = mx11 * x^2 + mx12 * x*y + mx13 * x*z + mx21 * x*y + mx22 * y^2 + …,

Let us start with the « revolution Mandelbrot », defined by mx11=1, mx22=mx33=-1, my12=my21=1 and mz13=mz31=1 (other coefficients = 0).



From there, we can slightly change the mij coeffcients, to see the effects on the shape. I will pass on all the coefficients that distort the shape in an « unfavourable » way (i.e. that clearly move us away from what a « 3D Mandelbrot » should be)
 
Here are some interesting results :
- changing my22, my23/my32, mz23/mz32, mz33 distorts the shape longitudinaly and creates a lateral « fractal spoke » in directions y and z. Watching from the front of the shape :



- changing my33 kind of « splits » the shape in the yz plane in an interesting way. Actually, when you combine the creation of a fractal spoke (on mz23/mz32) and this splitting (on my33), you get three spokes!
To make them identical and equally distributed around the x axis, the « best » parameters seem to be mz32=mz23=0,5 and my33=1/3 (or their opposites).




From there I tried to create additional spokes but could not do that just by changing the matrices coefficients. Since we want some kind of symmetry around the x axis, I tried to see how the iteration equations looked in the following polar coordinates :
x =  r cos ph
y = r cos th sin ph
z = r sin th sin ph

This gives us, in the iteration equations :
x = r^2 cos 2ph + cx
y = r^2 cos th sin 2ph + 1/6 r^2 sin^2 ph (cos 2th - 1) + cy
z = r^2 sin th cos 2ph + 1/2 r^2 sin^2 ph (sin 2th) + cz

The first coefficients are the ones of the « revolution Mandelbrot ».

The second coefficients in y and z equations are similar to a Mandelbrot equation in the yz plane, multiplied by a coefficient (r*sin ph) (i.e. r projected in the yz plane).
This gives the idea to choose a different power for th, and write :
x = r^p1 cos(p1*ph) + cx
y = r^p1 cos(th)sin(p1*ph) + 1/3 r^p1 sin(p1*ph)*cos(p2*th) + cy
z = r^ p1 sin(th)cos(p1*ph) + 1/2 r^p1 sin(p1*ph)*sin(p2*th) + cz

with p1 the power for ph, i.e. along x (p1=2 if you want a shape similar to the classical Mandelbrot)
and p2 the power for th, for the shape in the yz plane. It is as if you « combined » two Mandelbrot figures with different powers, one along x and the other along y and z.
When you increase p2, the number of spokes increases in a way similar to the way the 2D Mandelbrot evolves when you increase its power.
(Below for p2 = 3, 6, 20)










As you can see the p2=20 shape is quite hairy.

And yet,  still no sign of lateral spheres. Sorry Twinbee!

Excellent work FrozenOwl. Love the vectors and the matrices to clarify the kind of products involved.

Last weekend my friend made a small video of the fractal formula I wrote down.

It can be found at http://leden.vslcatena.nl/~robert/mandelbulb/Derickx-Equations.mp4